On some nonlinear dependence structure in portfolio design

Abstract

Constructing portfolios with high returns and low risks is always in great demand. Markowitz (1952) utilized correlation coefficients between pairs of stocks to build portfolios satisfying different levels of risk tolerance. The correlation coefficient describes the linear dependence structure between two stocks, but cannot capture a lot of nonlinear independence structures. Therefore, sometimes, portfolio performances are not up to investors' expectations. In this paper, based on the theory of copula by Sklar (see [19]), we investigate several new methods to detect nonlinear dependence structures. These new methods allow us to estimate the density of the portfolio which leads to calculations of some popular risk measurements like the value at risk (VaR) of investment portfolios. As for applications, making use of the listed stocks on the Ho Chi Minh city Stock Exchange (HoSE), some Markowitz optimal portfolios are constructed together with their risk measurements. Apparently, with nonlinear dependence structures, the risk evaluations of some pairs of stocks have noticeable twists. This, in turn, may lead to changes of decisions from investors.